Weighted norm inequalities of Hardy type for a class of integral operators. (English) Zbl 0837.26012

Let \(u\) and \(v\) be nonnegative measurable functions on \((0, \infty)\). Let \[ Kf(x)= \int^x_0 k(x, y) f(y) dy, \] where the kernel \(k\) satisfies the following conditions: \[ k(x, y)\geq 0\quad\text{for} \quad 0< y< x\tag{i} \] and it is non-decreasing in \(x\) or non-increasing in \(y\), \[ D^{- 1}(k(x, y)+ k(y, z))\leq k(x, z)\leq D(k(x, y)+ k(y, z))\quad\text{ if }\quad 0< z< y< x,\tag{ii} \] where \(D\geq 1\) is independent of \(x\), \(y\) and \(z\).
For the cases in which \(1< p,q< \infty\) necessary and sufficient conditions on \(u\) and \(v\) are found for the following inequality to hold with a constant \(C> 0\) independent of \(f\): \[ |(Kf) u|_q\leq C|fv|_p,\tag{1} \] where \(|\cdot|_s\) is the \(L_s\)- norm on \((0, \infty)\). Necessary and sufficient conditions of compactness of the operator \(K\) are obtained as well. If \(0< q< 1< p< \infty\) then necessary conditions and sufficient conditions for validity of (1) are also obtained.


26D10 Inequalities involving derivatives and differential and integral operators
42B99 Harmonic analysis in several variables
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